A119441 Distribution of A063834 in Abramowitz and Stegun order.

1, 2, 1, 3, 2, 1, 5, 3, 4, 2, 1, 7, 5, 6, 3, 4, 2, 1, 11, 7, 10, 9, 5, 6, 8, 3, 4, 2, 1, 15, 11, 14, 15, 7, 10, 9, 12, 5, 6, 8, 3, 4, 2, 1, 22, 15, 22, 21, 25, 11, 14, 15, 20, 18, 7, 10, 9, 12, 16, 5, 6, 8, 3, 4, 2, 1, 30, 22, 30, 33, 35, 15, 22, 21

Offset

1,2

Links

Table of n, a(n) for n=1..74.

M. Abramowitz and I. A. Stegun, eds., Handbook of Mathematical Functions, National Bureau of Standards, Applied Math. Series 55, Tenth Printing, 1972 [alternative scanned copy].

Formula

T(n,k) = product_{p=1..A036043(n,k)} A000041(c), 1<=k<=A000041(n), where c are the parts in the k-th partition of n. - R. J. Mathar, Jul 12 2013

Example

1;
2, 1;
3, 2, 1;
5, 3, 4, 2, 1;
7, 5, 6, 3, 4, 2, 1;
T(5,2) = 5 because the second partition of 5 is 1+4 and 4 can be repartitioned in 5 different ways.
T(5,3) = 6 because the third partition of 5 is 2+3, where the 2 can be partitioned in 2 ways (2, 1+1) and the 3 can be partitioned in 3 ways (3, 1+2, 1+1+1), 6=2*3.
T(5,4) = 3 because the fourth partition of 5 is 1+1+3 and 3 can be partitioned in 3 different ways.

Maple

# Compare two partitions (list) in AS order.
AScompare := proc(p1, p2)
    if nops(p1) > nops(p2) then
        return 1;
    elif nops(p1) < nops(p2) then
        return -1;
    else
        for i from 1 to nops(p1) do
            if op(i, p1) > op(i, p2) then
                return 1;
            elif op(i, p1) < op(i, p2) then
                return -1;
            end if;
        end do:
        return 0 ;
    end if;
end proc:
# Produce list of partitions in AS order
ASPrts := proc(n)
    local pi, insrt, p, ex ;
    pi := [] ;
    for p in combinat[partition](n) do
        insrt := 0 ;
        for ex from 1 to nops(pi) do
            if AScompare(p, op(ex, pi)) > 0 then
                insrt := ex ;
            end if;
        end do:
        if nops(pi) = 0 then
            pi := [p] ;
        elif insrt = 0 then
            pi := [p, op(pi)] ;
        elif insrt = nops(pi) then
            pi := [op(pi), p] ;
        else
            pi := [op(1..insrt, pi), p, op(insrt+1..nops(pi), pi)] ;
        end if;
    end do:
    return pi ;
end proc:
A119441 := proc(n, k)
    local pi, a, p ;
    pi := ASPrts(n)[k] ;
    a := 1 ;
    for p in pi do
        a := a*combinat[numbpart](p) ;
    end do:
    a ;
end proc:
for n from 1 to 10 do
    for k from 1 to A000041(n) do
        printf("%d, ", A119441(n, k)) ;
    end do:
    printf("\n") ;
end do: # R. J. Mathar, Jul 12 2013

Crossrefs

Cf. A063834, A119442, A000041 (row lengths and also first column)

Sequence in context: A304100 A179314 A204927 * A347227 A322083 A058399

Adjacent sequences: A119438 A119439 A119440 * A119442 A119443 A119444

Keyword

easy,nonn,tabf

Author

Alford Arnold, May 19 2006

Status

approved